Abstract
Relative equilibria occur in a wide variety of physical applications, including celestial mechanics, particle accelerators, plasma physics, and atomic physics. We derive sufficient conditions for Lyapunov stability of circular orbits in arbitrary axisymmetric gravitational (electrostatic) and magnetic fields, including the effects of local mass (charge) and current density. Particularly simple stability conditions are derived for source‐free regions, where the gravitational field is harmonic (∇2U = 0) or the magnetic field irrotational (∇ × B = 0). In either case the resulting stability conditions can be expressed geometrically (coordinate‐free) in terms of dimensionless stability indices. Stability bounds are calculated for several examples, including the problem of two fixed centers, the J2 planetary model, galactic disks, and a toroidal quadrupole magnetic field.
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Howard, J.E. Stability of relative equilibria in arbitrary axisymmetric gravitational and magnetic fields. Celestial Mechanics and Dynamical Astronomy 74, 19–57 (1999). https://doi.org/10.1023/A:1008388105585
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DOI: https://doi.org/10.1023/A:1008388105585